Waveform optimization technology of missile borne radar based on FSK-PSK composite modulation for anti repeater jamming

Zhiting FEI; Min ZHAO; Jindong ZHANG; Chen LI; Xiaohua ZHU

doi:10.1051/jnwpu/20224010125

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Volume 40 / No 1 (February 2022)

JNWPU, 40 1 (2022) 125-133

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Issue		JNWPU Volume 40, Number 1, February 2022


Page(s)		125 - 133
DOI		https://doi.org/10.1051/jnwpu/20224010125
Published online		02 May 2022

JNWPU 2022, 40(1): 125–133

Waveform optimization technology of missile borne radar based on FSK-PSK composite modulation for anti repeater jamming

基于FSK-PSK复合调制的弹载雷达抗转发干扰波形优化技术

Zhiting FEI (费智婷)¹^,2 (feizhiting@163.com), Min ZHAO (赵民)², Jindong ZHANG (张劲东)³, Chen LI (李晨)³ and Xiaohua ZHU (朱晓华)¹

¹ School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
² China Academy of Launch Vehicle Technology, Beijing 100076, China
³ College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received: 19 May 2021

Abstract

Aiming at the waveform of missile borne radar is simple and easy to be affected by the repeater jamming in the terminal guidance, an anti-jamming waveform optimization algorithm based on the frequency shift keying(FSK) and phase shift keying(PSK) composite modulation is proposed. The model function of FSK-PSK composite modulation signal is analyzed. Based on the orthogonality of sub pulses, an optimization algorithm of intra pulse PSK modulation based on the alternating direction multiplier method(ADMM) is designed. The simulation results show that comparing with PSK signal, the intra pulse sub pulse orthogonality of FSK-PSK signal optimized by modulation is improved by about 10 dB, which can effectively suppress the intra pulse forwarding interference.

摘要

针对弹载雷达波形易受末制导过程中转发干扰影响的问题, 提出了一种基于频移键控(FSK)和相移键控(PSK)复合调制的抗干扰波形优化算法。分析了FSK-PSK复合调制信号的模型函数, 在转发干扰模型基础上构建了脉内切片转发式干扰抑制模型, 基于子脉冲正交性设计了基于交替方向乘子法(ADMM)的脉内PSK调制优化算法。仿真结果表明, 与PSK信号相比, 调制优化的FSK-PSK信号脉内子脉冲正交性提升约10 dB, 能够有效抑制脉内转发干扰。

Key words: composite modulation signal / intrapulse slice repeater / waveform optimization / alternating direction method of multiplier

关键字 : 复合调制波形 / 脉内切片转发 / 波形优化 / 交替方向乘子法

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

电子对抗技术的迅速发展对末制导的探测能力提出了更高的要求[1]。目前弹载雷达的探测波形普遍较为简单，极易受到敌方的侦收和转发。因此，提高弹载雷达的波形复杂度，使其具备良好的低截获和抗干扰性能，具有重要的实际应用价值。

频移键控(FSK)和相移键控(PSK)的调制形式[2]目前普遍运用于各类雷达系统中，FSK-PSK的复合调制形式具有更高的复杂度，增加了干扰机截获的难度，具有更高的抗干扰能力。文献[3-4]从模糊函数、自相关函数等角度对FSK、PSK以及FSK-PSK复合调制信号进行了分析和比较，FSK-PSK信号的自相关函数主副瓣比优于FSK信号，并且编码序列长度越长，信号性能越优异。文献[5-7]从模糊函数、低截获性能等方面进行了分析。FSK-PSK信号比FSK和PSK信号具有更大的时宽带宽积和更佳的低截获性能。文献[8]利用Frank码调制相位编码信号、Costas码调制跳频编码信号，得到了一种新型FSK-PSK信号并对其进行较为细致的研究。文献[9-10]研究了混沌码与线性调频复合调制信号抗转发干扰的性能。

转发干扰，尤其是脉冲转发干扰[11]，使得导引头无法利用脉间的正交性进行干扰对抗。本文从转发干扰波形优化角度对FSK-PSK复合调制信号进行研究，在转发干扰模型基础上构建了脉内切片转发式干扰抑制模型，基于子脉冲正交性设计了基于交替方向乘子法(ADMM)的脉内PSK调制优化算法。

FSK-PSK复合调制信号是一种硬件上比较好实现的典型信号形式。相比线性调频信号，弹载雷达能够利用优化FSK-PSK波形进一步提升对转发干扰的抑制能力，因此具有重要的工程应用前景。

1 FSK-PSK复合调制雷达信号分析

1.1 FSK-PSK复合调制信号模型

FSK-PSK复合调制信号是通过脉内相位编码、脉间捷变频的方式进行调制的，即将一个宽脉冲分割为若干个彼此相接的子脉冲，利用跳频编码序列对子脉冲进行调制。然后利用相位编码序列对每个子脉冲进行调制，传统方法中每个子脉冲的相位编码序列是相同的，这样简单形式的信号已不足以应对现代日益发展的雷达电子对抗技术，因此，为了提高信号的复杂程度和信号脉内的隔离度，本文利用不同的相位编码序列对子脉冲进行调制。图 1显示了FSK-PSK信号的部分形式。

图中f_m表示第m个子脉冲的跳频, 若子脉冲的相位编码序列长度为N, 那么第m个子脉冲内的相位编码序列为, n=0, 1, …, N-1, 其中k_m (n)∈[0, 1, …, K-1], K表示离散相位个数。

设FSK-PSK复合调制信号的总脉冲宽度为T, 共有M个子脉冲, 那么子脉冲宽度T_c=T/M, 这样FSK-PSK复合调制信号可用(1)式表示

式中，v_m(t)为第m个子脉冲信号, 可写为

式中，, m=0, 1, …, M-1, σ_m ∈N₊为跳频系数, 而p_n (t)是宽度为的理想矩形脉冲。

相位编码信号的频谱表达式, 推导可得到[11]

因此, FSK-PSK信号的频谱表达式可写为

FSK-PSK信号带宽B_FSK-PSK与FSK信号带宽B_PSK相近, 即

若FSK-PSK信号中的跳频系数σ_m是一串可连续的正整数, 则

可得到FSK-PSK信号的模糊函数表达式

式中: m, l表示频点对应的子脉冲；(l-m-1)T_c≤τ≤(l-m+1)T_c, 且有

式中, n, r表示相位编码的对应序号

当|τ+(n-r)T_P+(m-l)T_c|≤T_P时, (9)式化简可得

图1

FSK-PSK信号波形示意图

1.2 FSK-PSK复合调制信号仿真分析

取脉冲宽度T=2 μs, 码元宽度T_P=0.1 μs, 采用跳频系数σ_m={1, 3, 2, 4}编码跳频编码序列, 相位编码序列等同于跳频编码序列，则FSK-PSK信号模糊函数如图 2所示。

令时延τ=0, FSK-PSK信号模糊函数可转换为距离模糊函数; 令多普勒频移ζ=0, FSK-PSK信号模糊函数转换为其速度模糊函数。图 3所示为图 2切割之后的距离、速度模糊函数图。

FSK-PSK信号在速度维的主峰幅度同样可直接利用速度模糊函数表示, 即

因此, FSK-PSK复合调制信号多普勒敏感, 其多普勒容限和单一的FSK信号和PSK信号是相同的。

图2

FSK-PSK信号模糊函数图

图3

FSK-PSK信号距离与速度模糊函数图

2 脉内切片转发式干扰抑制模型

2.1 脉内切片转发式模型

敌方干扰机产生脉内切片转发式干扰的时序模型如图 4所示[11]。干扰机在时域按照采样周期截获部分雷达信号, 生成截获信号, 可视为雷达信号与采样周期为T_P且宽度为τ₀的理想矩形脉冲相乘, 从而获得脉内切片信号。切片信号经功率放大、延时转发处理后生成脉内切片转发式干扰信号。雷达在接收到脉内转发切片干扰信号后, 由于转发干扰具有相参性, 形成多个欺骗假目标。

对于切片信号宽度等于FSK-PSK脉内子脉冲信号宽度的特定切片转发式干扰, 可利用脉内子脉冲正交波形来抑制[12-14]。

图4

切片转发式干扰模型示意图

2.2 脉内切片转发式干扰抑制模型

设一个码元内采样N_S点, 则采样周期为T_S=T_P/N_S, 单个子脉冲信号内总采样点数N′=N·N_S。将(2)式重写为

式中: 表示采样后的第m个子脉冲信号的相位编码序列; p′_n(t)表示宽度为T_S的理想矩形脉冲。

将子脉冲信号表达式(13)代入相关函数定义式

取τ=kT_S, -(N′-1)≤k≤N′-1, 则(14)式可转变为

将χ_ml(kT_P) 简写为χ_ml(k), 定义

那么FSK-PSK脉内子脉冲信号的互相关函数的离散形式可表示为

当m=l时, (19)式表示子脉冲信号的自相关函数的离散形式

式中, 。

为了提高FSK-PSK脉内子脉冲信号的正交性, 可将优化模型描述为: 以脉内子脉冲信号的自相关函数副瓣电平、互相关函数电平加权和的最小化为准则, 固定跳频编码序列以离散相位编码序列为优化变量, 对FSK-PSK信号进行优化设计。目标函数可表示为

式中: Ψ=[k|N_S≤|k|≤N′-1]表示自相关的副瓣范围; Φ=[k|0≤|k|≤N′-1]表示互相关函数范围; α∈[0, 1]为加权系数, 当α=1时, 目标函数只考虑抑制自相关函数的副瓣, 而α=0时, 目标函数只考虑抑制互相关函数。

3 抗转发干扰波形优化

3.1 ADMM算法

ADMM算法解决的是2个变量下的优化问题, 它的方法框架结构是可分的, 通过分解原优化问题交替求解2个较易解决的子问题的方式, 就可以减小问题的规模[15]。在求解过程中, 将l₁范数中的变量进行替换形成l₁+l₂的形式, 然后逐次求解, 使问题得到解决。文献[16-20]体现该方法在雷达波形优化方面降低复杂度的优势。

ADMM框架的基本模型为

将(22)式写成增广拉格朗日函数, ρ为惩罚项系数, 有

则ADMM更新公式为

设置一个较小的正数δ作为收敛门限, 当‖ x^(t+1)- z^(t+1)‖ < δ时即认为原问题收敛。

3.2 基于ADMM的抗转发切片波形优化算法

利用ADMM架构嵌套拟牛顿法优化FSK-PSK脉内子脉冲信号的自相关函数副瓣电平、互相关函数电平, 使两者加权和最小化。优化问题为

(25) 式可转换为

在问题(26)中引入辅助变量z和约束条件c= z, 即

问题(26)和问题(27)等价。根据(27)式写出增广拉格朗日方程

记u=(λ_r+jλ_i)/ρ, 则(28)式可写为

记c^(t)为第t次ADMM迭代后的c值(z和u相应的记为z^(t), u^(t)), 给定初值c⁽⁰⁾, z⁽⁰⁾, u⁽⁰⁾则该问题的求解可按照如下步骤进行:

1) 更新变量c, 此时变量z^(t), u^(t)被看作已知量

该问题为无约束最优化问题, 可使用拟牛顿法求解, 首先需将复数形式的目标函数转化为实数形式, 设

则有

在后续计算中, 要将实向量转化为, 还原成复向量c_m^(t+1)。

2) 更新变量z, 此时变量c^(t+1), u(t)被看作已知量

(33) 式可转化为

由于(34)式是一个线性问题, 且z中的元素彼此独立, 则可将其分解为若干个子问题

若将复数c_m^(t+1)(n)+ u_m^(t)(n), z_m^(t)(n)看成复平面上的向量, 则(35)式中目标函数可表示成

忽略常值部分, 则(35)式转化为

即求整数k_m(n)(0≤k_m(n) < K), 使向量z_m^(t)(n)和[c_m^(t+1)(n)+ u_m^(t)(n)]的夹角最小, k_m(n)的解为

则(37)式的解为

3) 更新变量u, 此时其他变量被看作已知量

基于ADMM的抗转发切片波形优化算法的求解步骤可总结为: 初始化变量c⁽⁰⁾, z⁽⁰⁾, u⁽⁰⁾, 然后重复步骤1)到步骤3), 当迭代次数达到设置的最大迭代次数, 或者|δ^(t+1)-δ^(t)|≤ε时算法终止, 其中ε是一个较小正数, 表示目标函数变化值达到收敛门限。

3.3 计算复杂度分析

在上述ADMM过程中, (30)式占据了大部分的计算量和计算过程, 由于其需要计算(30)式的梯度。考虑到FSK-PSK的优化调制序列个数为MN, 因此其梯度的计算量包含复数乘法次数为

因此, 基于ADMM的抗转发切片波形优化算法的计算复杂度达到了Ο(M⁵N⁵)。

总体来看, 虽然该算法计算复杂度优于遗传算法等搜索类算法, 但整体复杂度仍较高。该算法适合非在线计算。由于实际电子对抗环境下转发模型参数的先验信息可以通过雷达提前侦测或干扰分析等手段获取, 因此通过提前计算的方式可以对FSK-PSK波形进行优化。

4 仿真校验

设计FSK-PSK信号的子脉冲数目M=4, 子脉冲信号的跳频系数分别对应了σ_m={1, 3, 2, 4}, 子脉冲信号的相位编码码长N=50, 离散相位数K=2。设置优化模型的加权系数分别为α={0, 0.1, …, 1}, 优化算法迭代次数上限为50次, 分别用上述ADMM算法以及GA算法对相同的初始波形进行脉内子脉冲信号正交波形优化。其优化前后结果对比图如图 5所示。

根据基于FSK-PSK信号的脉内子脉冲正交波形设计模型, 定义脉内子脉冲信号的自相关-平均副瓣电平(auto correlation-average sidelobe level, AC-ASL)、互相关-平均电平(cross correlation-average level, CC-AL)

式中, Ψ=[k|N_S≤|k|≤N′-1]是自相关函数的副瓣区域, 而Φ=[k|0≤|k|≤N′-1]则是信号互相关区域。

由图 5a)可知, 随着加权系数的增加, 对AC-ASL的优化比重增加, 经过2种算法优化后的AC-ASL值在总体趋势上逐渐下降。GA算法相比ADMM算法对AC-ASL的抑制更加明显, 其值都在未优化波形和ADMM优化波形AC-ASL值之下。而ADMM算法优化的波形在加权系数较小时, 其AC-ASL值高于未优化波形AC-ASL, 在加权系数较大时, 其AC-ASL值才低于未优化波形AC-ASL值。由图 5b)可知, 随着加权系数的增加, 对CC-AL的优化比重逐渐减小, 经过2种算法优化的CC-AL值在总体趋势上逐渐上升。经过GA算法优化之后的CC-AL值均大于未优化前波形和ADMM算法优化波形的CC-AL值。

从图 6~7可以发现, 经过ADMM算法优化之后的FSK-PSK脉内子脉冲信号自相关函数副瓣以及互相关函数均得到抑制。表 1所示为经过ADMM算法优化前后的结果对比表。由表 1可知, 经过ADMM算法优化后, 波形AC-ASL和CC-AL均得到抑制。随着离散相位数K的增加, 优化效果未必进一步提升。

对比FSK-PSK信号和PSK信号的AC-ASL与CC-AL，具体数据如表 2所示。由表 2可知，FSK-PSK信号的AC-ASL值等同于PSK信号的AC-ASL值，但是FSK-PSK信号的CC-AL值明显优于PSK信号的CC-AL值。与PSK信号相比，FSK-PSK对转发干扰的改善接近10 dB。

图5

优化前后AC-ASL与CC-AL结果对比

图6

优化前后FSK-PSK脉内子脉冲信号自相关函数对比

图7

优化前后FSK-PSK脉内子脉冲信号互相关函数对比

表1

优化前后AC-ASL和CC-AL数据对比 dB

表2

FSK-PSK信号和PSK信号对比 dB

5 结论

针对干扰机产生的特定长度脉内切片转发式干扰，提出了基于FSK-PSK信号的脉内等长子脉冲正交波形对抗该切片转发式干扰。通过基于ADMM的算法实现对相关函数的联合优化，实现对FSK-PSK信号自相关函数旁瓣电平与互相关函数电平的抑制。仿真结果表明，不同于GA算法以牺牲互相关函数的代价来换取对自相关函数副瓣的抑制，此算法可在部分加权系数之下实现对FSK-PSK脉内子脉冲信号的自相关函数和互相关函数的联合优化。优化后的FSK-PSK信号可实现子脉冲的高正交性，可有效抑制特定脉内切片转发式干扰。与PSK信号相比，FSK-PSK对转发干扰的改善接近10 dB。

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Cheng Z, Liao B, He Z. A nonlinear-ADMM method for designing mimo radar constant modulus waveform with low correlation sidelobes[J]. Signal Processing, 2019, 15993–103. [Article] [CrossRef] [Google Scholar]
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Tang B, Li J, Liang J. Alternating direction method of multipliers for radar waveform design in spectrally crowded environments[J]. Signal Processing, 2018, 142398–402. [Article] [CrossRef] [Google Scholar]
Wang Y, Wang J. Constant modulus probing waveform design for mimo radar via admm algorithm[C]//2018 IEEE International Conference on Acoustics, Speech and Signal Processing, 2018 [Google Scholar]
Raei E, Alaee-Kerahroodi M, Shankar M R B. ADMM based transmit waveform and receive filter design in cognitive radar systems[C]//2020 IEEE Radar Conference, 2020 [Google Scholar]

All Tables

表1

优化前后AC-ASL和CC-AL数据对比 dB

In the text

表2

FSK-PSK信号和PSK信号对比 dB

In the text

All Figures

	图1 FSK-PSK信号波形示意图
In the text

	图2 FSK-PSK信号模糊函数图
In the text

	图3 FSK-PSK信号距离与速度模糊函数图
In the text

	图4 切片转发式干扰模型示意图
In the text

	图5 优化前后AC-ASL与CC-AL结果对比
In the text

	图6 优化前后FSK-PSK脉内子脉冲信号自相关函数对比
In the text

	图7 优化前后FSK-PSK脉内子脉冲信号互相关函数对比
In the text

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